5
\end{array} \right) + \lambda \left( \begin{array} { l }
1
1
3
\end{array} \right) + \mu \left( \begin{array} { c }
- 1
2
1
\end{array} \right) .$$
- Find a vector which is perpendicular to \(\Pi\).
- Hence find an equation for \(\Pi\) in the form r.n \(= p\).
- Find in the form \(\sqrt { q }\) the shortest distance between \(\Pi\) and the origin, where \(q\) is a rational number.
4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c r c } a & 2 & 3
4 & 4 & 6
- 2 & 2 & 9 \end{array} \right)\) where \(a\) is a constant. It is given that if \(\mathbf { A }\) is not singular then
$$\mathbf { A } ^ { - 1 } = \frac { 1 } { 24 a - 48 } \left( \begin{array} { c c c }
24 & - 12 & 0
- 48 & 9 a + 6 & 12 - 6 a
16 & - 2 a - 4 & 4 a - 8
\end{array} \right)$$ - Use \(\mathbf { A } ^ { - 1 }\) to solve the simultaneous equations below, giving your answer in terms of \(k\).
$$\begin{array} { r }
x + 2 y + 3 z = 6
4 x + 4 y + 6 z = 8
- 2 x + 2 y + 9 z = k
\end{array}$$ - Consider the equations below where \(a\) takes the value which makes \(\mathbf { A }\) singular.
$$\begin{aligned}
a x + 2 y + 3 z & = b
4 x + 4 y + 6 z & = 10
- 2 x + 2 y + 9 z & = - 13
\end{aligned}$$
\(b\) takes the value for which the equations have an infinite number of solutions.
- Determine the value of \(b\).
- Find the solutions for \(y\) and \(z\) in terms of \(x\).
- For the equations in part (ii) with the values of \(a\) and \(b\) found in part (ii) describe fully the geometrical arrangement of the planes represented by the equations.
5 The region \(R\) between the \(x\)-axis, the curve \(y = \frac { 1 } { \sqrt { p + x ^ { 2 } } }\) and the lines \(x = \sqrt { p }\) and \(x = \sqrt { 3 p }\), where \(p\) is a positive parameter, is rotated by \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution \(S\). - Find and simplify an algebraic expression, in terms of \(p\), for the exact volume of \(S\).
- Given that \(R\) must lie entirely between the lines \(x = 1\) and \(x = \sqrt { 48 }\) find in exact form
- the greatest possible value of the volume of \(S\)
- the least possible value of the volume of \(S\).